Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Connect with a community of experts ready to provide precise solutions to your questions quickly and accurately. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To find the points of intersection of the line joining the points [tex]\((2, 4, 5)\)[/tex] and [tex]\((3, 5, -4)\)[/tex] with the specified planes, we need to examine the line in parametric form and solve for the parameter [tex]\( t \)[/tex] where the line intersects each plane.
Step-by-step solution:
1. First, determine the direction vector of the line:
[tex]\[ \text{Direction vector} = (3 - 2, 5 - 4, -4 - 5) = (1, 1, -9) \][/tex]
2. Express the line in parametric form:
[tex]\[ (x, y, z) = (2, 4, 5) + t \cdot (1, 1, -9) \][/tex]
So,
[tex]\[ x = 2 + t, \quad y = 4 + t, \quad z = 5 - 9t \][/tex]
### (a) Intersection with the xy-plane (where [tex]\( z = 0 \)[/tex])
To find the intersection with the xy-plane, set [tex]\( z = 0 \)[/tex]:
[tex]\[ 5 - 9t = 0 \implies t = \frac{5}{9} \][/tex]
Substitute [tex]\( t = \frac{5}{9} \)[/tex] back into the parametric equations for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = 2 + \frac{5}{9} = \frac{18}{9} + \frac{5}{9} = \frac{23}{9} \approx 2.56 \][/tex]
[tex]\[ y = 4 + \frac{5}{9} = \frac{36}{9} + \frac{5}{9} = \frac{41}{9} \approx 4.56 \][/tex]
Thus, the point of intersection with the xy-plane is approximately:
[tex]\[ \left(\frac{23}{9}, \frac{41}{9}, 0\right) \approx (2.56, 4.56, 0) \][/tex]
### (b) Intersection with the yz-plane (where [tex]\( x = 0 \)[/tex])
To find the intersection with the yz-plane, set [tex]\( x = 0 \)[/tex]:
[tex]\[ 2 + t = 0 \implies t = -2 \][/tex]
Substitute [tex]\( t = -2 \)[/tex] back into the parametric equations for [tex]\( y \)[/tex] and [tex]\( z \)[/tex]:
[tex]\[ y = 4 + (-2) = 2 \][/tex]
[tex]\[ z = 5 - 9(-2) = 5 + 18 = 23 \][/tex]
Thus, the point of intersection with the yz-plane is:
[tex]\[ (0, 2, 23) \][/tex]
### (c) Intersection with the zx-plane (where [tex]\( y = 0 \)[/tex])
To find the intersection with the zx-plane, set [tex]\( y = 0 \)[/tex]:
[tex]\[ 4 + t = 0 \implies t = -4 \][/tex]
Substitute [tex]\( t = -4 \)[/tex] back into the parametric equations for [tex]\( x \)[/tex] and [tex]\( z \)[/tex]:
[tex]\[ x = 2 + (-4) = -2 \][/tex]
[tex]\[ z = 5 - 9(-4) = 5 + 36 = 41 \][/tex]
Thus, the point of intersection with the zx-plane is:
[tex]\[ (-2, 0, 41) \][/tex]
### Summary of Intersection Points
1. The intersection with the xy-plane is:
[tex]\[ \left(\frac{23}{9}, \frac{41}{9}, 0\right) \approx (2.56, 4.56, 0) \][/tex]
2. The intersection with the yz-plane is:
[tex]\[ (0, 2, 23) \][/tex]
3. The intersection with the zx-plane is:
[tex]\[ (-2, 0, 41) \][/tex]
Step-by-step solution:
1. First, determine the direction vector of the line:
[tex]\[ \text{Direction vector} = (3 - 2, 5 - 4, -4 - 5) = (1, 1, -9) \][/tex]
2. Express the line in parametric form:
[tex]\[ (x, y, z) = (2, 4, 5) + t \cdot (1, 1, -9) \][/tex]
So,
[tex]\[ x = 2 + t, \quad y = 4 + t, \quad z = 5 - 9t \][/tex]
### (a) Intersection with the xy-plane (where [tex]\( z = 0 \)[/tex])
To find the intersection with the xy-plane, set [tex]\( z = 0 \)[/tex]:
[tex]\[ 5 - 9t = 0 \implies t = \frac{5}{9} \][/tex]
Substitute [tex]\( t = \frac{5}{9} \)[/tex] back into the parametric equations for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = 2 + \frac{5}{9} = \frac{18}{9} + \frac{5}{9} = \frac{23}{9} \approx 2.56 \][/tex]
[tex]\[ y = 4 + \frac{5}{9} = \frac{36}{9} + \frac{5}{9} = \frac{41}{9} \approx 4.56 \][/tex]
Thus, the point of intersection with the xy-plane is approximately:
[tex]\[ \left(\frac{23}{9}, \frac{41}{9}, 0\right) \approx (2.56, 4.56, 0) \][/tex]
### (b) Intersection with the yz-plane (where [tex]\( x = 0 \)[/tex])
To find the intersection with the yz-plane, set [tex]\( x = 0 \)[/tex]:
[tex]\[ 2 + t = 0 \implies t = -2 \][/tex]
Substitute [tex]\( t = -2 \)[/tex] back into the parametric equations for [tex]\( y \)[/tex] and [tex]\( z \)[/tex]:
[tex]\[ y = 4 + (-2) = 2 \][/tex]
[tex]\[ z = 5 - 9(-2) = 5 + 18 = 23 \][/tex]
Thus, the point of intersection with the yz-plane is:
[tex]\[ (0, 2, 23) \][/tex]
### (c) Intersection with the zx-plane (where [tex]\( y = 0 \)[/tex])
To find the intersection with the zx-plane, set [tex]\( y = 0 \)[/tex]:
[tex]\[ 4 + t = 0 \implies t = -4 \][/tex]
Substitute [tex]\( t = -4 \)[/tex] back into the parametric equations for [tex]\( x \)[/tex] and [tex]\( z \)[/tex]:
[tex]\[ x = 2 + (-4) = -2 \][/tex]
[tex]\[ z = 5 - 9(-4) = 5 + 36 = 41 \][/tex]
Thus, the point of intersection with the zx-plane is:
[tex]\[ (-2, 0, 41) \][/tex]
### Summary of Intersection Points
1. The intersection with the xy-plane is:
[tex]\[ \left(\frac{23}{9}, \frac{41}{9}, 0\right) \approx (2.56, 4.56, 0) \][/tex]
2. The intersection with the yz-plane is:
[tex]\[ (0, 2, 23) \][/tex]
3. The intersection with the zx-plane is:
[tex]\[ (-2, 0, 41) \][/tex]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.